The asymptotic Plateau problem for Hypersurfaces of constant $H_k$ curvature in hyperbolic space

In this paper, we study the asymptotic Plateau problem in hyperbolic space for hypersurfaces of constant $H_k$-curvature. We prove the existence of a smooth complete $k$-convex hypersurface in ${\mathbb{H}}^{n+1}$ satisfying $$H_k(\kappa )=\sigma ,\sigma \in (0,1),$$ with prescribed asymptotic boundary at infinity. In particular, our result extends the range of the constant $\sigma$ in the existence theorem of Guan and Spruck [J. Eur. Math. Soc. 12 (2010), no. 3, 797–817] for $H_k$ curvature to the full interval $(0,1)$.